Which transformations preserve size — and which don't.
The Mnemonic
The Size-Preserving Three
An isometry is any transformation that preserves distance — meaning the distance between any two points in the pre-image is EXACTLY the same as the distance between their corresponding points in the image. Translations, reflections, and rotations are all isometries.
Dilation is the one basic transformation that is NOT an isometry (except in the trivial case of scale factor k=1) — since it deliberately changes distances by the scale factor k, it fails the very definition of distance-preservation.
Same dimensions, same shape, every time — isometries never stretch, shrink, or distort
💡 Memory Trick
"Which transformations preserve size — and which don't." ISO- means "equal" (like isosceles = equal legs) — an ISOmetry keeps measurements EQUAL between pre-image and image. Translation, Reflection, Rotation: all isometries (the image is always congruent to the pre-image). Dilation: the odd one out, since it's the only transformation designed to change size.
Why It Works
Why This Matters for Congruence Proofs
Since isometries preserve every distance, they also automatically preserve every angle (angles are determined entirely by the relative positions and distances of points), and therefore preserve the entire shape's structure. This means the image produced by any isometry is always CONGRUENT to the pre-image — connecting directly back to the very first lesson of this course, CPCTC, and to Congruence Shortcuts.
In fact, an alternative, transformation-based definition of congruence exists: two figures are congruent if and only if one can be mapped onto the other using some composition of isometries (translations, reflections, and rotations). This is a genuinely different way of defining congruence than the SSS/SAS/ASA/AAS/HL shortcuts — both definitions describe the exact same underlying idea, just from different angles.
Using It In A Proof
Using Isometries to Prove Congruence
A transformation-based congruence proof works differently from a traditional two-column proof — instead of citing SSS/SAS/etc., you demonstrate a specific sequence of isometries that maps one figure exactly onto the other.
1
Identify a sequence of isometries mapping one shape to the other
This might be a single transformation, or a composition of several (translation + reflection, for example).
2
Verify each individual transformation used is genuinely an isometry
Confirm no dilation (or any size-changing step) is included in the sequence — including one would break the congruence claim entirely.
3
Conclude congruence
If a composition of ONLY isometries maps the pre-image exactly onto the image, the two figures are congruent by definition.
Full Worked Example
Proving Congruence Using a Composition of Isometries
Given: Triangle A has vertices (1,1), (4,1), (1,3). Triangle B has vertices (−1,−1), (−4,−1), (−1,−3). Prove: Triangle A is congruent to Triangle B using transformations.
1
Compare the coordinates
Every coordinate in Triangle B is exactly the negative of the corresponding coordinate in Triangle A: (1,1)→(−1,−1), (4,1)→(−4,−1), (1,3)→(−1,−3).
2
Match this pattern to a known rule
(x,y)→(−x,−y) is exactly the 180° rotation rule about the origin.
3
Confirm this is an isometry
Rotation is one of the three isometries (translation, reflection, rotation) — no dilation or size change is involved.
4
Conclude congruence
Since a single isometry (180° rotation about the origin) maps Triangle A exactly onto Triangle B, the two triangles are congruent.
This is a genuinely different proof METHOD than citing SSS or SAS directly, even though it reaches the same congruence conclusion — transformations give geometry a second valid path to the same kind of result.
🎯 Quick Worked Example
A shape undergoes a translation, then a dilation with scale factor 3. Is the final image congruent to the original pre-image?
1
Check each transformation in the sequence. Translation is an isometry (preserves distance).
2
Check the second transformation. Dilation with scale factor 3 changes every distance by a factor of 3 — it is NOT an isometry.
3
Conclude. Since the sequence includes a non-isometry (the dilation), the final image is NOT congruent to the original — it's similar instead (same shape, different size).
📌 Exam Application
A composition that includes even ONE non-isometry (any dilation with k≠1) breaks congruence for the entire sequence, no matter how many isometries are also included — always check every single step in a composition, since one size-changing transformation is enough to disqualify a congruence claim.
⚠️ Most Common Isometries Mistakes
Trap 1 — Assuming any transformation preserves congruence: Only translations, reflections, and rotations (isometries) preserve congruence — dilation with any scale factor other than 1 produces a similar, not congruent, image.
Trap 2 — Forgetting that a composition needs EVERY step to be an isometry: If a composition combines an isometry with a dilation, the overall result is not congruent, even though part of the sequence technically preserved distance — the presence of any single non-isometry step breaks the congruence conclusion for the whole composition.
✓ Quick Self-Test
1) What does 'isometry' mean? 2) Which three basic transformations are isometries? 3) Why is dilation (with k≠1) not an isometry? 4) How does the transformation-based definition of congruence relate to CPCTC and Congruence Shortcuts? 5) A shape is reflected, then rotated, then dilated by k=1 — is the final image congruent to the original? Why or why not?